Percolation thresholds on a triangular lattice for neighborhoods containing sites up to the fifth coordination zone

We determine thresholds p_c for random-site percolation on a triangular lattice for all available neighborhoods containing sites from the first to the fifth coordination zones, including their complex combinations. There are 31 distinct neighborhoods. The dependence of the value of the percolation thresholds p_c on the coordination number z are tested against various theoretical predictions. The proposed single scalar index ξ =∑_iz_ir_i²/i (depending on the coordination zone number i , the neighborhood coordination number z , and the square distance r² to sites in i th coordination zone from the central site) allows one to differentiate among various neighborhoods and relate p_c to ξ . The thresholds roughly follow a power law p_c∝ξ^−γ with γ ≈0.710 (19 ) .